A birth, death, and diffusion process

Human Evolutionary Trees.

“…(iti) Adke and Moyal (1963) provide a result similar to (3.4.4) for unlabelled histories descended from a stngle population but no direct proof is given.…”

Some generalizations of Bailey's birth death and migration model

“…However, an iterative form of solution can be given in terms of the c.fl. 's for the process conditional upon the population size at time t being N(t) = s, along the lines of Adke and Moyal ((1963), Sections II and IV). Substituting the probability of zero survivors at time t, while for s~1:…”

“…A model equivalent to Bailey's is obtained by taking this Markov process to be the simple symmetrical random walk on the lattice points. Another case of the general model has been con-sidered by Adke and Moyal (1963), who assumed that the population size was determined by the simple birth and death process, and that the individuals were subject to ordinary Brownian diffusion along the real line. These authors discussed the asymptotic distribution of mean position and spatial dispersion of the population, conditional upon a fixed finite number of survivors.…”

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

“…To understand what we mean by BBM being borderline, it is useful to consider BBM as a special case of a class of Gaussian processes labelled by a function A : [0, 1] → [0, 1] with A(0) = 0, A(1) = 1 which is increasing and rightcontinuous. Given such a function, so-called variable speed branching Brownian motion [20,21,31,11,12] can then be constructed in two equivalent ways 1 .…”